A collection of MATLAB classes for computing and using spherical harmonic transforms is presented. Methods of these classes compute differential operators on the sphere and are used to solve simple partial differential equations in a spherical geometry. The spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions are presented in detail. A set of methods associated with a spectral_field class provides spectral approximation to the differential operators ∇ ⋯, ∇ ×, ∇, and ∇ 2 in spherical geometry. Laplace inversion and Helmholtz equation solvers are also methods for this class. The use of the class and methods in MATLAB is demonstrated by the solution of the barotropic vorticity equation on the sphere. A survey of alternative algorithms is given and implementations for parallel high performance computers are discussed in the context of global climate and weather models.

The Parallel Community Climate Model (PCCM) is a message-passing parallelization of version 2.1 of the Community Climate Model (CCM) developed by researchers at Argonne and Oak Ridge National Laboratories and at the National Center for Atmospheric Research in the early to mid 1990s. In preparation for use in the Department of Energy’s Parallel Climate Model (PCM), PCCM has recently been updated with new physics routines from version 3.2 of the CCM, improvements to the parallel implementation, and ports to the SGI/Cray Research T3E and Origin 2000. We describe our experience in porting and tuning PCCM on these new platforms, evaluating the performance of different parallel algorithm options and comparing performance between the T3E and Origin 2000. 1

The numerical methods used to solve the time-dependent Dirac equation on a three-dimensional Cartesian lattice are described. Efficient algorithms are required for computationally intensive studies of nonperturbative relativistic quantum dynamics. Discretization is achieved through the lattice basis-spline collocation method, in which quantum-state vectors and coordinate-space operators are expressed in terms of basis-spline functions on a spatial lattice. All numerical procedures reduce to a series of matrix-vector operations which are performed on the Intel iPSC/860 hypercube, making full use of parallelism. Solutions to the problems of limited node memory and node-to-node communication overhead inherent in using distributed-memory, multiple-instruction, multiple-data stream parallel computers are discussed.

One approach to improving the real-time efficiency of plasma turbulence calculations is to use a parallel algorithm. A parallel algorithm for plasma turbulence calculations was tested on the Intel iPSC/860 hypercube and the Touchstone Delta machine. Using the 128 processors of the Intel iPSC/860 hypercube, a factor of 5 improvement over a single-processor CRAY-2 is obtained. For the Touchstone Delta machine, the corresponding improvement factor is 13. For plasma edge turbulence calculations, an extrapolation of the present results to the Intel sigma machine gives an improvement factor close to 64 over the single-processor CRAY-2.>

Abstract The spectral transform method is a widely used numerical technique for solving partial differential equations on the sphere in global climate modeling. This paper describes the parallelization and performance of the spectral method for solving the non‐linear shallow water equations on the surface of a sphere using a 128‐node Intel iPSC/860 hypercube. Solving the shallow water equations represents a computational kernel of more complex climate models. This work is part of a research program to develop climate models that are capable of much longer simulations at a significantly finer resolution than current models. Such models are important in understanding the effects of the increasing atmospheric concentrations of greenhouse gases, and the computational requirements are so large that massively parallel multiprocessors will be necessary to run climate model simulations in a reasonable amount of time. The spectral method involves the transformation of data between the physical, Fourier and spectral domains. Each of these domains is two‐dimensional. The spectral method performs Fourier transforms in the longitude direction followed by summation in the latitude direction to evaluate the discrete spectral transform. A simple way of parallelizing the spectral code is to decompose the physical problem domain in just the latitude direction. This allows an optimized sequential FFT algorithm to be used in the longitude direction. However, this approach limits the number of processors that can be brought to bear on the problem. Decomposing the problem over both directions allows the parallelism inherent in the problem to be exploited more effectively‐the grain size is reduced, so that more processors can be used. Results are presented that show that decomposing over both directions does result in a more rapid solution of the problem. The results show that, for a given problem and number of processors, the optimum decomposition has approximately equal numbers of processors in each direction. Load imbalance also has an impact on the performance of the method. The importance of minimizing communication latency and overlapping communication with calculation is stressed. General methods for doing this, that may be applied to many other problems, are discussed.

Abstract The spectral transform method is a standard numerical technique used to solve partial differential equations on the sphere in global climate modeling. In particular, it is used in CCM1 and CCM2, the Community Climate Models developed at the National Center for Atmospheric Research. This paper describes initial experiences in parallelizing a program that uses the spectral transform method to solve the non‐linear shallow water equations on the sphere, showing that an efficient implementation is possible on the Intel iPSC/860. The use of PICL, a portable instrumented communication library, and Paragraph, a performance visualization tool, in tuning the implementation is also described. The Legendre transform and the Fourier transform comprise the computational kernel of the spectral transform method. This paper is a case study of parallelizing the Legendre transform. For many problem sizes and numbers of processors, the spectral transform method can be parallelized efficiently by parallelizing only the Legendre transform.

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